Properties

Label 2.64.ax_jh
Base Field $\F_{2^{6}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{6}}$
Dimension:  $2$
L-polynomial:  $1 - 23 x + 241 x^{2} - 1472 x^{3} + 4096 x^{4}$
Frobenius angles:  $\pm0.0377720538760$, $\pm0.353369870209$
Angle rank:  $2$ (numerical)
Number field:  4.0.4369673.1
Galois group:  $D_{4}$
Jacobians:  12

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2843 16583219 68731094336 281389648536803 1152810846348255963 4722305717453457894656 19342799144632178136552107 79228165098340015170714572483 324518555408363527208533796455232 1329227994595395976355885887897719219

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 42 4050 262191 16772130 1073638762 68718592479 4398043334874 281474985891138 18014398606622991 1152921503575102930

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The endomorphism algebra of this simple isogeny class is 4.0.4369673.1.
All geometric endomorphisms are defined over $\F_{2^{6}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.x_jh$2$(not in LMFDB)